Semiparametric estimation of spectral density function for irregular spatial data

Estimation of the covariance structure of spatial processes is of fundamental importance in spatial statistics. In the literature, several non-parametric and semi-parametric methods have been developed to estimate the covariance structure based on the spectral representation of covariance functions. However,they either ignore the high frequency properties of the spectral density, which are essential to determine the performance of interpolation procedures such as Kriging, or lack of theoretical justification. We propose a new semi-parametric method to estimate spectral densities of isotropic spatial processes with irregular observations. The spectral density function at low frequencies is estimated using smoothing spline, while a parametric model is used for the spectral density at high frequencies, and the parameters are estimated by a method-of-moment approach based on empirical variograms at small lags. We derive the asymptotic bounds for bias and variance of the proposed estimator. The simulation study shows that our method outperforms the existing non-parametric estimator by several performance criteria.Comment: 29 pages, 2 figure

Spatial Multiresolution Cluster Detection Method

A novel multi-resolution cluster detection (MCD) method is proposed to identify irregularly shaped clusters in space. Multi-scale test statistic on a single cell is derived based on likelihood ratio statistic for Bernoulli sequence, Poisson sequence and Normal sequence. A neighborhood variability measure is defined to select the optimal test threshold. The MCD method is compared with single scale testing methods controlling for false discovery rate and the spatial scan statistics using simulation and f-MRI data. The MCD method is shown to be more effective for discovering irregularly shaped clusters, and the implementation of this method does not require heavy computation, making it suitable for cluster detection for large spatial data

Compressed Distributed Gradient Descent: Communication-Efficient Consensus over Networks

Network consensus optimization has received increasing attention in recent years and has found important applications in many scientific and engineering fields. To solve network consensus optimization problems, one of the most well-known approaches is the distributed gradient descent method (DGD). However, in networks with slow communication rates, DGD's performance is unsatisfactory for solving high-dimensional network consensus problems due to the communication bottleneck. This motivates us to design a communication-efficient DGD-type algorithm based on compressed information exchanges. Our contributions in this paper are three-fold: i) We develop a communication-efficient algorithm called amplified-differential compression DGD (ADC-DGD) and show that it converges under {\em any} unbiased compression operator; ii) We rigorously prove the convergence performances of ADC-DGD and show that they match with those of DGD without compression; iii) We reveal an interesting phase transition phenomenon in the convergence speed of ADC-DGD. Collectively, our findings advance the state-of-the-art of network consensus optimization theory.Comment: 11 pages, 11 figures, IEEE INFOCOM 201

Spatial CUSUM for Signal Region Detection

Detecting weak clustered signal in spatial data is important but challenging in applications such as medical image and epidemiology. A more efficient detection algorithm can provide more precise early warning, and effectively reduce the decision risk and cost. To date, many methods have been developed to detect signals with spatial structures. However, most of the existing methods are either too conservative for weak signals or computationally too intensive. In this paper, we consider a novel method named Spatial CUSUM (SCUSUM), which employs the idea of the CUSUM procedure and false discovery rate controlling. We develop theoretical properties of the method which indicates that asymptotically SCUSUM can reach high classification accuracy. In the simulation study, we demonstrate that SCUSUM is sensitive to weak spatial signals. This new method is applied to a real fMRI dataset as illustration, and more irregular weak spatial signals are detected in the images compared to some existing methods, including the conventional FDR, FDR$_L$ and scan statistics

Variance Estimation and Kriging Prediction for a Class of Non-Stationary Spatial Models

This paper discusses the estimation and plug-in kriging prediction non-stationary spatial process assuming a smoothly varying variance an additive independent measurement error. A difference-based kernel estimator of the variance function and a modified likelihood estimator of the mea surement error variance are used for parameter estimation. Asymptotic properties of these estimators and the plug-in kriging predictor are established. A simula tion study is presented to test our estimation-prediction procedure. Our kriging predictor is shown to perform better than the spatial adaptive local polynomial regression estimator proposed by Fan and Gijbels (1995) when the measurement error is small

Estimation and Prediction of a Class of Convolution-Based Spatial Nonstationary Models for Large Spatial Data

In this article we address two important issues common to the analysis of large spatial datasets. One is the modeling of nonstationarity, and the other is the computational challenges in doing likelihood-based estimation and kriging prediction. We model the spatial process as a convolution of independent Gaussian processes, with the spatially varying kernel function given by the modified Bessel functions. This is a generalization of the process-convolution approach of Higdon, Swall, and Kern (1999), who used the Gaussian kernel to obtain a closed-form nonstationary covariance function. Our model can produce processes with richer local behavior similar to the processes with the Matérn class of covariance functions. Because the covariance function of our model does not have a closed-form expression, direct estimation and spatial prediction using kriging is infeasible for large datasets. Efficient algorithms for parameter estimation and spatial prediction are proposed and implemented. We compare our method with methods based on stationary model and moving window kriging. Simulation results and application to a rainfall dataset show that our method has better prediction performance. Supplemental materials for the article are available online

Variance function estimation of a one-dimensional nonstationary process

We propose a flexible nonparametric estimation of a variance function from a one-dimensional process where the process errors are nonstationary and correlated. Due to nonstationarity a local variogram is defined, and its asymptotic properties are derived. We include a bandwidth selection method for smoothing taking into account the correlations in the errors. We compare the proposed difference-based nonparametric approach with Anderes and Stein(2011)’s local-likelihood approach. Our method has a smaller integrated MSE, easily fixes the boundary bias, and requires far less computing time than the likelihood-based method

Estimating spatial covariance using penalised likelihood with weighted L1 penalty

In spatial statistics, the estimation of covariance matrices is of great importance because of its role in spatial prediction and design. In this paper, we propose a penalised likelihood approach with weighted L 1 regularisation to estimate the covariance matrix for spatial Gaussian Markov random field models with unspecified neighbourhood structures. A new algorithm for ordering spatial points is proposed such that the corresponding precision matrix can be estimated more effectively. Furthermore, we develop an efficient algorithm to minimise the penalised likelihood via a novel usage of the regularised solution path algorithm, which does not require the use of iterative algorithms. By exploiting the sparsity structure in the precision matrix, we show that the LASSO type of approach gives improved covariance estimators measured by several criteria. Asymptotic properties of our proposed estimator are derived. Both our simulated examples and an application to the rainfall data set show that the proposed method performs competitively